The paper examines the effect of conditional heteroskedasticity on least squares inference in stochastic regression models of unknown integration order and proposes an inference procedure that is robust to models within the (near) I(0)–(near) I(1) range with GARCH innovations. We show that a regressor signal of exact order $O_{p}\left ( n\kappa _{n}\right ) $ for arbitrary $\,\kappa _{n}\rightarrow \infty $ is sufficient to eliminate stationary GARCH effects from the limit distributions of least squares based estimators and self-normalized test statistics. The above order dominates the $O_{p}\left ( n\right ) $ signal of stationary regressors but may be dominated by the $O_{p}\left ( n^{2}\right ) $ signal of I(1) regressors, thereby showing that least squares invariance to GARCH effects is not an exclusively I(1) phenomenon but extends to processes with persistence degree arbitrarily close to stationarity. The theory validates standard inference for self normalized test statistics based on the ordinary least squares estimator when $\kappa _{n}\rightarrow \infty $ and $\kappa _{n}/n\rightarrow 0$ and the IVX estimator (Phillips and Magdalinos (2009a), Econometric Inference in the Vicinity of Unity. Working paper, Singapore Management University; Kostakis, Magdalinos, and Stamatogiannis, 2015a, Review of Financial Studies 28(5), 1506–1553.) when $\kappa _{n}\rightarrow \infty $ and the innovation sequence of the system is a covariance stationary vec-GARCH process. An adjusted version of the IVX–Wald test is shown to also accommodate GARCH effects in purely stationary regressors, thereby extending the procedure’s validity over the entire (near) I(0)–(near) I(1) range of regressors under conditional heteroskedasticity in the innovations. It is hoped that the wide range of applicability of this adjusted IVX–Wald test, established in Theorem 4.4, presents an advantage for the procedure’s suitability as a tool for applied research.

本文检验了条件异方差性对未知积分阶数的随机回归模型中最小二乘推理的影响，并提出了一种推理过程，该过程对（近）I（0）–（近）I（1）范围内的模型具有 GARCH 鲁棒性创新。我们表明， 对于任意 $\,\kappa _{n}\rightarrow \infty $ ，精确阶 $O_{p}\left ( n\kappa _{n}\right ) $ 的回归量信号足以消除平稳 GARCH基于最小二乘法的估计量和自归一化检验统计量的极限分布的影响。上面的顺序支配了 $O_{p}\left ( n\right ) $ 平稳回归量的信号，但可能被 $O_{p}\left ( n^{2}\right ) $ 支配I(1) 回归变量的信号，从而表明 GARCH 效应的最小二乘不变性不是唯一的 I(1) 现象，而是扩展到持久性程度任意接近平稳性的过程。 该理论基于$\kappa _{n}\rightarrow \infty $ 和 $\kappa _{n}/n\rightarrow 0$ 时的普通最小二乘估计量 和 IVX 估计量（Phillips和 Magdalinos (2009a), Econometric Inference in the Vicinity of Unity. Working paper, Singapore Management University; Kostakis, Magdalinos, and Stamatogiannis, 2015a, Review of Financial Studies 28(5), 1506–1553.) when $\kappa _{ n}\rightarrow \infty $ 系统的新息序列是一个协方差平稳的vec-GARCH过程。IVX-Wald 检验的调整版本显示也适用于纯稳态回归变量中的 GARCH 效应，从而将程序的有效性扩展到条件异方差下整个（近）I(0)–（近）I(1) 回归量范围在创新中。希望在定理 4.4 中建立的这种调整后的 IVX-Wald 检验的广泛适用性为该程序作为应用研究工具的适用性提供了优势。